1Institute of Mathematics The Hebrew University Jerusalem, Israel and Department of Mathematics Rutgers University New Brunswick, NJ 08903, U.S.A. 2Department of Mathematics P.O. Box 4 00014 University of Helsinki, Finland
Fundamenta Mathematicae, Tome 174 (2002) no. 2, pp. 97-126
For a cardinal $\kappa$ and a model $M$ of cardinality $\kappa$ let
${\rm No}(M)$ denote the number of nonisomorphic
models of cardinality $\kappa$ which are
$L_{\infty,\kappa}$-equivalent to $M$.
We prove that for $\kappa$ a weakly compact cardinal, the question of
the possible values of ${\rm No}(M)$ for models $M$ of
cardinality $\kappa$ is equivalent to the question of the possible
numbers of equivalence classes of equivalence relations which are
$\Sigma^1_1$-definable over $V_\kappa$.
By \cite {ShVa719} it is possible to have a generic extension where
the possible numbers of equivalence classes of
$\Sigma^1_1$-equivalence relations are in a prearranged set.
Together these results settle the problem of the possible values of
${\rm No}(M)$ for models of weakly compact
cardinality.
Keywords:
cardinal kappa model cardinality kappa denote number nonisomorphic models cardinality kappa which infty kappa equivalent prove kappa weakly compact cardinal question possible values models cardinality kappa equivalent question possible numbers equivalence classes equivalence relations which sigma definable kappa cite shva possible have generic extension where possible numbers equivalence classes sigma equivalence relations prearranged set together these results settle problem possible values models weakly compact cardinality
Affiliations des auteurs :
Saharon Shelah
1
;
Pauli Vaisanen
2
1
Institute of Mathematics The Hebrew University Jerusalem, Israel and Department of Mathematics Rutgers University New Brunswick, NJ 08903, U.S.A.
2
Department of Mathematics P.O. Box 4 00014 University of Helsinki, Finland
Saharon Shelah; Pauli Vaisanen. The number of $L_{\infty\kappa}$-equivalent
nonisomorphic models for $\kappa$ weakly compact. Fundamenta Mathematicae, Tome 174 (2002) no. 2, pp. 97-126. doi: 10.4064/fm174-2-1
@article{10_4064_fm174_2_1,
author = {Saharon Shelah and Pauli Vaisanen},
title = {The number of $L_{\infty\kappa}$-equivalent
nonisomorphic models for $\kappa$ weakly compact},
journal = {Fundamenta Mathematicae},
pages = {97--126},
year = {2002},
volume = {174},
number = {2},
doi = {10.4064/fm174-2-1},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4064/fm174-2-1/}
}
TY - JOUR
AU - Saharon Shelah
AU - Pauli Vaisanen
TI - The number of $L_{\infty\kappa}$-equivalent
nonisomorphic models for $\kappa$ weakly compact
JO - Fundamenta Mathematicae
PY - 2002
SP - 97
EP - 126
VL - 174
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.4064/fm174-2-1/
DO - 10.4064/fm174-2-1
LA - en
ID - 10_4064_fm174_2_1
ER -
%0 Journal Article
%A Saharon Shelah
%A Pauli Vaisanen
%T The number of $L_{\infty\kappa}$-equivalent
nonisomorphic models for $\kappa$ weakly compact
%J Fundamenta Mathematicae
%D 2002
%P 97-126
%V 174
%N 2
%U http://geodesic.mathdoc.fr/articles/10.4064/fm174-2-1/
%R 10.4064/fm174-2-1
%G en
%F 10_4064_fm174_2_1
